Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C24 | C4×D4 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,14)(11,12)(15,16)(17,18,19,21,20,22,23,24);(2,4)(9,10)(11,15)(12,14)(13,16);(6,8)(10,14)(11,12)(15,16)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,f^b=fh,d^c=dh,f^c=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72321116, #72321139, #72321144, … |
| maximal normal subgroup of | #72321116, #72321139, #72321144, … |
| maximal quotient of | #72322389 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18414);